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Let $D$ be a Carathéodory domain. For $1\le p\le \infty$, let $L^p\left(D\right)$ be the class of all functions $f$ holomorphic in $D$ such that ${\parallel f\parallel }_{D,p}=[\frac{1}{A}\int {\int }_D^{}{\left|f\left(z\right)\right|}^p{dxdy]}^{1/p}<\infty$, where $A$ is the area of $D$. For $f\in L^p\left(D\right)$, set $$E_n^p\left(f\right)=\underset{t\in {\pi }_n}{inf}{\parallel f-t\parallel }_{D,p};$$ ${\pi }_n$ consists of all polynomials of degree at most $n$. In this paper we study the growth of an entire function in terms of approximation error in $L^p$-norm on $D$.